5.1 Ground state crust
The EoS of the outer crust in the ground state approximation (see Section 3.1) is rather well
established, so that the pressure at any given density is determined within a few percent accuracy
[42, 183, 357] as can be seen in Figure 27.
||EoS of the ground state of the outer crust for various nuclear models. From Rüster et
al. . A zoomed-in segment of the EoS just before the neutron drip can be seen in Figure 28.
||EoS of the ground state of the outer crust just before neutron drip for various nuclear
models. From Rüster et al. .
Uncertainties arise above the density as shown in Figure 28 because
experimental data are lacking. However, one may hope that with the improvement of experimental
techniques, experimental data on very exotic nuclei will become available in the future.
On the contrary, the inner crust nuclei cannot be studied in a laboratory because their
properties are influenced by the gas of dripped neutrons, as reviewed in Section 3.2. This means
that only theoretical models can be used there and consequently the EoS after neutron drip is
much more uncertain than in the outer layers. The neutron gas contributes more and more
to the total pressure with increasing density. Therefore, the problem of correct modeling of
the EoS of a pure neutron gas at subnuclear densities becomes important. The true EoS of
cold catalyzed matter stems from a true nucleon Hamiltonian, expected to describe nucleon
interactions at , where is the nuclear saturation density. To make the solution of the
many-body problem feasible, the task is reduced to finding an effective nucleon Hamiltonian, which
would enable one to calculate reliably both the properties of laboratory nuclei and the EoS of
cold catalyzed matter for . The task also includes the calculation of the
crust-core transition. We will illustrate the general results with two examples of the EoS of
the inner crust, calculated in the compressible–liquid-drop model (see Section 3.2.1) using the
effective nucleon-nucleon interactions FPS (Friedman–Panharipande–Skyrme ) and SLy
(Skyrme–Lyon [86, 88, 87]).
||Comparison of the SLy and FPS EoSs. From .
As one can see in Figure 29, significant differences between the SLy and FPS EoS are restricted to the
density interval 4 1011 – 4 1012 g cm–3. They result mainly from the fact that the density at
which neutron drip occurs for each is different: (in good agreement
with the “empirical EoS” of Haensel & Pichon ), while .
For the SLy and FPS EoSs are very similar, with the
FPS EoS being a little softer at the highest densities considered. The detailed behavior of the
two EoSs near the crust-core transition can be seen in Figure 30. The FPS EoS is softer there
than the SLy EoS (for pure neutron matter the FPS model is softer at subnuclear densities;
||Comparison of the SLy and FPS EoSs near the crust-core transition. Thick solid line:
inner crust with spherical nuclei. Dashed line corresponds to “exotic nuclear shapes”. Thin solid line:
uniform npe matter. From .
In the case of the SLy EoS, the crust-liquid core transition takes place as a very weak first-order phase
transition, with a relative density jump on the order of one percent. Notice that, for this model, spherical
nuclei persist to the very bottom of the crust . As seen from Figure 30, the crust-core transition is
accompanied by a noticeable stiffening of the EoS. For the FPS EoS the situation is different. Namely,
the crust-core transition takes place through a sequence of phase transitions with changes of
nuclear shapes as discussed in Section 3.3. These phase transitions make the crust-core transition
smoother than in the SLy case, with a gradual increase of stiffness (see Figure 35). While the
presence of exotic nuclear shapes is expected to have dramatic consequences for the transport,
neutrino emission, and elastic properties of neutron star matter, their effect on the EoS is rather
||Adiabatic index for various EoSs of the ground-state outer crust below
neutron drip. The horizontal line corresponds to . The neutron drip point
4 1011 g cm–3 depends slightly on the EoS model used and, therefore, is not marked.
From Rüster et al. .
||Adiabatic index for the EoS of the ground-state crust. Dotted vertical lines correspond
to the neutron drip and crust-core interface points. Calculations performed using the SLy EoS of
Douchin & Haensel .
||The SLy EoS. Dotted vertical lines correspond to the neutron drip and crust-core
transition. From .
The overall SLy EoS of the crust, calculated including adjacent segments of the liquid core and the outer
crust, is shown in Figure 33. In the outer crust segment, the SLy EoS cannot be visually distinguished from
the EoSs of Haensel & Pichon  or Rüster et al. , which are based on experimental nuclear
masses. An important dimensionless parameter, measuring the stiffness of an EoS at a given density, is the
which at subnuclear density can be approximated by . The total pressure is defined
by Equation (24). The adiabatic index is shown in Figures 31 and 32 as a function of mass density
. At , we have . This is because in these outer crust layers, pressure is
very well approximated by the sum of the contribution of the ultrarelativistic electron gas () and of
the lattice contribution (), which both have the same density dependence (see
Section 3.1). One notices in Figure 32 a dramatic softening in the density region following
the neutron drip point, . This means, in particular, that no stable stars can exist
with central densities around because the compressibility of the matter is too low. Then
the EoS stiffens gradually, with a significant increase of near the crust-core interface. A
jump in on the core side is connected with the disappearance of nuclei, and a subsequent
stiffening (due to the nucleon-nucleon interaction) in the uniform liquid. Figures 31 and 32
show that the EoS of the inner crust is very different from the polytropic form ,
where is a constant. Tabulated and analytical EoSs of the ground state crust are available
online [210, 118].